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# Archimedean solid

An Archimedean solid or semiregular solid is a convex polyhedron with regular polygons as faces, such that at least two different types of regular polygons are used, and all vertices are identical (in the sense that the polygons are arranged in the same way about each vertex, and if someone rotates an Archimedean solid with no markings randomly, when you aren't looking, it is not possible to work out any information at all about which vertex was moved where.). For short: they are vertex-uniform but not face-uniform. The regular-faced prisms and antiprisms are also semiregular and fit this criteria, but for historical reasons are not included in the definition (probably because there are infinitely many of them, or because of their lesser symmetry). The elongated square gyrobicupola (a Johnson solid) appears at first to meet the above criteria, but upon closer inspection it is not vertex-uniform at all, the vertices being only locally similar. Compare to Platonic solids, which are face-uniform (in addition to being vertex-uniform), and to Johnson solids, which need not be vertex-uniform.

The Archimedean solids are known to have been discussed by Archimedes, although the complete record is lost. During the Renaissance, artists and mathematicians valued pure forms and rediscovered all of these forms. This search culminated in the work of Johannes Kepler circa 1619, who defined prisms, antiprisms, and the non-convex solids known as Kepler-Poinsot solids.

All edges of an Archimedean solid have the same length, since the faces are regular polygons, and the edges of a regular polygon have the same length. The neighbours of a polygon must have the same edge length, therefore also the neighbours of the neighbours, and so on.

There are 13 Archimedean solids (15 if the mirror images of two enantiomorphs, see below, are counted separately).

Name and picture Faces Edges Vertices Faces meeting at each vertex Symmetry group
cuboctahedron

(Video)
14 (8 triangles, 6 squares) 24 12 triangle-square-triangle-square Oh
icosidodecahedron

(Video)
32 (20 triangles, 12 pentagons) 60 30 triangle-pentagon-triangle-pentagon Ih
truncated tetrahedron

(Video)
8 (4 triangles, 4 hexagons) 18 12 triangle-hexagon-hexagon Td
truncated cube or truncated hexahedron

(Video)
14 (8 triangles, 6 octagons) 36 24 triangle-octagon-octagon Oh
truncated octahedron

(Video)
14 (6 squares, 8 hexagons) 36 24 square-hexagon-hexagon Oh
truncated dodecahedron

(Video)
32 (20 triangles, 12 decagons) 90 60 triangle-decagon-decagon Ih
truncated icosahedron or commonly football (soccer ball)

(Video)
32 (12 pentagons, 20 hexagons) 90 60 pentagon-hexagon-hexagon Ih
rhombicuboctahedron or small rhombicuboctahedron

(Video)
26 (8 triangles, 18 squares) 48 24 triangle-square-square-square Oh
truncated cuboctahedron or great rhombicuboctahedron

(Video)
26 (12 squares, 8 hexagons, 6 octagons) 72 48 square-hexagon-octagon Oh
rhombicosidodecahedron or small rhombicosidodecahedron

(Video)
62 (20 triangles, 30 squares, 12 pentagons) 120 60 triangle-square-pentagon-square Ih
truncated icosidodecahedron or great rhombicosidodecahedron

(Video)
62 (30 squares, 20 hexagons, 12 decagons) 180 120 square-hexagon-decagon Ih
snub cube or snub cuboctahedron (2 chiral forms)

(Video)

(Video)
38 (32 triangles, 6 squares) 60 24 triangle-triangle-triangle-triangle-square O
snub dodecahedron or snub icosidodecahedron (2 chiral forms)

(Video)

(Video)
92 (80 triangles, 12 pentagons) 150 60 triangle-triangle-triangle-triangle-pentagon I

The first two solids (cuboctahedron and icosidodecahedron) are edge-uniform and are called quasi-regular.

The last two (snub cube and snub dodecahedron) are known as chiral, as they come in a left-handed (Latin: levomorph or laevomorph) form and right-handed (Latin: dextromorph) form. When something comes in multiple forms which are each other's three-dimensional mirror image, these forms may be called enantiomorphs. (This nomenclature is also used for the forms of chemical compounds).

The duals of the Archimedean solids are called the Catalan solids. Together with the bipyramids and trapezohedra, these are the face-uniform solids with regular vertices.